The number of pairs of real numbers $(x, y)$ such that $x = x^2 + y^2$ and $y = 2xy$ is:

Let $\alpha$ and $\beta$ be the roots of $x^{2}-3x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^{2}-6x+q=0$. If $\alpha, \beta, \gamma, \delta$ form a geometric progression,then the ratio $(2q+p):(2q-p)$ is:

If $x = \frac{1}{2} \left( \sqrt{7} + \frac{1}{\sqrt{7}} \right)$,then the value of $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to

If the equation $x^8 - kx^2 + 3 = 0$ has a real solution,then the least integral value of $k$ is-

If the equation whose roots are $p$ times the roots of the equation $x^4-2ax^3+4bx^2+8ax+16=0$ is a reciprocal equation,then $|p|=$ :

Which of the following is incorrect? If $a \equiv b \pmod{m}$ and $x$ is an integer,then